News from Mathnasium of Cedar Park

The History of Zero

Apr 6, 2016

How was zero discovered?

Nils-Bertil Wallin

YaleGlobal, 19 November 2002

The phenomenon of zero.

From placeholder to the driver of calculus, zero has crossed the greatest minds and most diverse borders since it was born many centuries ago. Today, zero is perhaps the most pervasive global symbol known. In the story of zero, something can be made out of nothing.

Zero, zip, zilch - how often has a question been answered by one of these words? Countless, no doubt. Yet behind this seemingly simple answer conveying nothing lays the story of an idea that took many centuries to develop, many countries to cross, and many minds to comprehend. Understanding and working with zero is the basis of our world today; without zero we would lack calculus, financial accounting, the ability to make arithmetic computations quickly, and, especially in today's connected world, computers. The story of zero is the story of an idea that has aroused the imagination of great minds across the globe.

When anyone thinks of one hundred, two hundred, or seven thousand the image in his or her mind is of a digit followed by a few zeros. The zero functions as a placeholder; that is, three zeroes denotes that there are seven thousands, rather than only seven hundreds. If we were missing one zero, that would drastically change the amount. Just imagine having one zero erased (or added) to your salary! Yet, the number system we use today - Arabic, though it in fact came originally from India - is relatively new. For centuries people marked quantities with a variety of symbols and figures, although it was awkward to perform the simplest arithmetic calculations with these number systems.

The Sumerians were the first to develop a counting system to keep an account of their stock of goods - cattle, horses, and donkeys, for example. The Sumerian system was positional; that is, the placement of a particular symbol relative to others denoted its value. The Sumerian system was handed down to the Akkadians around 2500 BC and then to the Babylonians in 2000 BC. It was the Babylonians who first conceived of a mark to signify that a number was absent from a column; just as 0 in 1025 signifies that there are no hundreds in that number. Although zero's Babylonian ancestor was a good start, it would still be centuries before the symbol as we know it appeared.

The renowned mathematicians among the Ancient Greeks, who learned the fundamentals of their math from the Egyptians, did not have a name for zero, nor did their system feature a placeholder as did the Babylonian. They may have pondered it, but there is no conclusive evidence to say the symbol even existed in their language. It was the Indians who began to understand zero both as a symbol and as an idea.

Brahmagupta, around 650 AD, was the first to formalize arithmetic operations using zero. He used dots underneath numbers to indicate a zero. These dots were alternately referred to as 'sunya', which means empty, or 'kha', which means place. Brahmagupta wrote standard rules for reaching zero through addition and subtraction as well as the results of operations with zero. The only error in his rules was division by zero, which would have to wait for Isaac Newton and G.W. Leibniz to tackle.

But it would still be a few centuries before zero reached Europe. First, the great Arabian voyagers would bring the texts of Brahmagupta and his colleagues back from India along with spices and other exotic items. Zero reached Baghdad by 773 AD and would be developed in the Middle East by Arabian mathematicians who would base their numbers on the Indian system. In the ninth century, Mohammed ibn-Musa al-Khowarizmi was the first to work on equations that equaled zero, or algebra as it has come to be known. He also developed quick methods for multiplying and dividing numbers known as algorithms (a corruption of his name). Al-Khowarizmi called zero 'sifr', from which our cipher is derived. By 879 AD, zero was written almost as we now know it, an oval - but in this case smaller than the other numbers. And thanks to the conquest of Spain by the Moors, zero finally reached Europe; by the middle of the twelfth century, translations of Al-Khowarizmi's work had weaved their way to England.

The Italian mathematician, Fibonacci, built on Al-Khowarizmi's work with algorithms in his book Liber Abaci, or "Abacus book," in 1202. Until that time, the abacus had been the most prevalent tool to perform arithmetic operations. Fibonacci's developments quickly gained notice by Italian merchants and German bankers, especially the use of zero. Accountants knew their books were balanced when the positive and negative amounts of their assets and liabilities equaled zero. But governments were still suspicious of Arabic numerals because of the ease in which it was possible to change one symbol into another. Though outlawed, merchants continued to use zero in encrypted messages, thus the derivation of the word cipher, meaning code, from the Arabic sifr.

The next great mathematician to use zero was Rene Descartes, the founder of the Cartesian coordinate system. As anyone who has had to graph a triangle or a parabola knows, Descartes' origin is (0,0). Although zero was now becoming more common, the developers of calculus, Newton and Lebiniz, would make the final step in understanding zero.

Adding, subtracting, and multiplying by zero are relatively simple operations. But division by zero has confused even great minds. How many times does zero go into ten? Or, how many non-existent apples go into two apples? The answer is indeterminate, but working with this concept is the key to calculus. For example, when one drives to the store, the speed of the car is never constant - stoplights, traffic jams, and different speed limits all cause the car to speed up or slow down. But how would one find the speed of the car at one particular instant? This is where zero and calculus enter the picture.

If you wanted to know your speed at a particular instant, you would have to measure the change in speed that occurs over a set period of time. By making that set period smaller and smaller, you could reasonably estimate the speed at that instant. In effect, as you make the change in time approach zero, the ratio of the change in speed to the change in time becomes similar to some number over zero - the same problem that stumped Brahmagupta.

In the 1600's, Newton and Leibniz solved this problem independently and opened the world to tremendous possibilities. By working with numbers as they approach zero, calculus was born without which we wouldn't have physics, engineering, and many aspects of economics and finance.

In the twenty-first century zero is so familiar that to talk about it seems like much ado about nothing. But it is precisely understanding and working with this nothing that has allowed civilization to progress. The development of zero across continents, centuries, and minds has made it one of the greatest accomplishments of human society. Because math is a global language, and calculus its crowning achievement, zero exists and is used everywhere. But, like its function as a symbol and a concept meant to denote absence, zero may still seem like nothing at all. Yet, recall the fears over Y2K and zero no longer seems like a tale told by an idiot.

References:
1. Kaplan, Robert (2000). The Nothing that Is: A Natural History of Zero. New York: Oxford University Press.

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